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Iterative

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Iterative

    Iterative

    JournalofZhejiangUniversitySCIENCEA

    ISSN1673565X(Print);ISSN18621775(Online)

    wwwzjueducn/jzus;wwwspringerlinkeom

    Email:JZUS@ZjUeducn

    Huoetal/JZhejiangUnivSciA20078(121..19211927

    Iterativelearningcon

    ARXidentifi

    trol0fSoFCbasedon

    cationmodel

    HUOHai.bo,ZHUXin-jian,TUHengyong

    (FuelCellRes'earchInstitute,Depaltnlent(),Automation.ShanghaiJiaoTongUniversiO,,Shanghai200240,China)

    E-mail:huohbl028sjmedu.cD

    ReceixedJuly30,2007;rexisionacceptedOct8,2007

    1921

    Abstract:ThispaperpresentsallapplicationofiterativelearningcontrolfILC1techniquetothevoltagecontrolofsolidoxide

    thelcell(SOFC)stack.Tomeetthedemandsofthecontrolsystemdesign,anautoregressivemodelwithexogenousinput(ARX)is

    established.Firstly.byregulatingthevariationofthehydrogenflowrateproportionaltothatofthecurrentthefuelutilizationof

    theSOFCiskeptwithinitsadmissiblerange.ThenbasedontheARXmodel,threekindsofILCcontrollersi.e.P,Pland

    PD

    typearedesignedtokeepthevoltageatadesiredleve1.SimulationresultsdemonstratethepotentialoftheARXmodelapplied

tothecontroloftheSOFCandprovetheexcellenceoftheILCcontrollersforthevoltagecontro

    loftheSOFC.

    Keywords:Autoregressivemodelwithexogenousinput(ARX),lterativelearningcontrol(I

    LC),Solidoxidefuelcell(SOFC)

    Identication

    doi:10.1631/jzus.2007A1921Documentcode:ACLCnumber:TP273;TM911.4

    lNTRODUCTlON

    Distributedgeneration(DG)isapl'omising technologythatcanbeusedtoaddresssomeofthe technica1aswel1asenvironmenta1concernsinpower systems.Asakindofhightemperaturerue1cel1.solid

    oxidefue1cel1fSOFC1presentsanattractiveoption f0rtheDGtechnologybecauseitjsmodular,emcient andenvironmentallyfriendly.Unlikeothertypesof rue1cells.SOFCisentirelysolidstatewithno1iquid components.Anditusuallyworksatahightern. perature.intherangeof800-1000.Ctoreachthe electrolytesionicconductivityrequirement(Qiela1., 2005).

    SOFCisadynamicdexicewhichwil1affectthe dynamicbehaviorofthepowersystemtowhichitis connected.Inordertoanalyzesuchbehavior.accu

    ratedynamicmodelsoftheSOFCarerequired.Unti1 now,severa1nonlineardynamicmodelsoftheSOFC havebeenproposed(HallandColclaser,1999;Pa

    Project(No2006AA05Z148)supportedbytheHiTechResearch

    andDevelopmentProgram(863)ofChina

    dulldsela1..2000:ZhuandTomsovic.2002: SedghisigarchiandFeliachi2004;Kandepuela1., 2007;Udagawaela1.,2007).Mostofthesemodels

    emphasizethedetaileddescriptionofcel1interna1 processessuchascomponentmaterialbalance,en

    ergybalanceandelectrochemica1kinetics.etc.These modelsareveryusefu1toanalyzethetransientchar. acteristicsoftheSOFC.buttheyaretoocomplicated t0beusedforcontro1systemdesign.Tomeetthe demandsofdevelopingvalidcontro1strategiesJu. rado(2004;2006)presentedtheidentificationmodels oftheSOFC.Butoneofthemostimportantcel1per. formancevariables,fue1utilizationhasnotbeen examinedwhentheyexploredtheSOFCdynamic responseafterdisturbances.

    Systemidentificationisaprocesswhichcon. structsamathematica1mode1byinputandoutputdata f-0radynamicsystemundertesting.ItisusuallyaD. pliedinthecasewhentheoretica1modelingistoo complex.AlthoughthenonlinearnatureofanSOFC mustberecognized.inmanycasesa1inearizedsys

    ternrepresentationallowsforamoreecientmeans

    ofanalysis.Byfar.anautoregressivemode1with

1922Huoeta1./JZhefiangUnivSciA20078(12J:19211927

    exogenousinput(ARX)isthemostwidelyapplied lineardynamicmodel(Ghaffarieta1..2007).Anditis alsoamongthemostcommonandrobustmodeltypes usedbyengineerstocharacterizelinearsystems (Ljung.1999).Sointhispaper.anARxmodelofthe SOFCsuitedforcontrolsystemdesignisestablished firstly.inwhichoperatingissuesaboutthefuelutili. zationareconsideredspecifically.

Iterativelearningcontrol(ILC),firstlyintro

    ducedbyArimotoeta1.(1984)iswellknownwithits abtVtodetermineacontrolinputiterativelysothat thetrackingofagivenreferencesignalortheoutput trajectoryoverafixedtimeintervalispossible.Until nowILChasbeensuccessfullyappliedinmany fieldssuchasroboticmanipulators,chemicalbatch processes.etc.However.theconcretestudy0fILC schemesdesignfortheSOFChasnotbeenfoundin priorliterature.

    SOFCgeneratesDCpowerfromruelandoxi

    dantviaanelectrochemicalprocess.Anychangesin theloadcircuitoritspowerdemandcausethe changesoftheS0FCoperatingconditions.Asaresult. theoutputvoltageoftheSOFCwillhaveahigh fluctuation.Furthermore.ifthesechangesresultinthe fueloftheSOFCtobeoverusedorunderused.theFC willbedamagedpermanently.Inthisstudy,thefuel utilizationoftheSOFCiskeptwithinitsadmissible rangebyregulatingthevariationofthehydrogenflow rateproportionaltothatofthecurrent.Theninorder tokeeptheDCvoltageatadesiredleve1.ILC schemesbasedontheARXmodelaredeveloped. THE0RYF0RS0FCDYNAMICM0DEL

    Inthefollowings,someknowledgeaboutthe nonlineardynamicmodeloftheSOFCandfuel utilizationisbrieflyintroduced.

    SOFCdynamicmodel

    ThenonlineardynamicmodeloftheSOFCstack adoptedinthispaperisshowninFig.1(Padull6seta1.,

2000).Thismodelisbasedonthefollowingassump

    tions:

    (1)Thegasesareidea1.

    (2)Stackisfedwithhydrogenandoxygen(fuel processordynamicsisnotincluded).

    f31Thetemperatureisstableatal1time. q:

    

    Fig.1Dynamicmodelofthesolidoxidefuelcell (SOFC1stack

    (41Theratioofpressuresbetweentheinterior andexteriorofthechannelislargeenoughtoconsider theorificechoked(Padull6setaL,2000;Sedghisi

    garchiandFeliachi.2004).

    DCcurrentintheelectricalcircuitacrosstwo electrodesisgeneratedduetothereleasedelectrons (2)attheanode.ApplyingNernst'sequationand Ohm'slaw(takingintoaccountohmiclosses).the stackoutputvoltagecouldbemodeledasfollows (Padull6seta1.2000;Lieta1.2007):

    =

    .+-nPH,p2)

    ,

    where

    ?:(懿一2KrI),,

    ():(g-KrI),

    ):,,1U,

    (2)

    (3)

    (4)

where,Eo

    ,

    TandParestackcurrent,standard

    reversiblecellpotential,stackoutputvoltage,oper- atingtemperatureandpartialpressureoftheithre

    actant,respectively.qLandg0in,areinputhydrogen flowandoxygenflow,respectively;KH, ,Ko,

    and

    KH,

    0arevalvemolarconstantsforhydrogen,oxygen andwateLrespectively;,

    ,To

    ,

    andTHoarere

    sponsetimesforhydrogenflOW,oxygenflowand waterflOW,respectively;isohmicresistance.

Huoetal/JZh~iangUnivSciA20078(12.'19217927

    Fuplutilization

    Fuelutilizationisoneofthemostimportant variablesaffectingtheperformanceofFC,whichis definedas

    

    gJ,

    /e/=————一=—一=_

    gg2In

    ForprotectingSOFCstack.thedesiredrangeoffuel utilizationissetbetween0.7and0.9. Inordertokeepthefuelutilizationwithinits

admissiblerangeandtoprotectthestack,thefo1

    lowingconstraintisimposedonutilization (SedghisigarchiandFeliachi,2006) where

    A"=0.

    Imax,

    Au={0,

    lmIf1,

    11>"max,

    m_n

    <<11max,

    11</e/mIn.

    BylinearizingEq.(5),itbecomes

    2Fqi~I一器2,(8)2F(g"

    where,J0isnominalstackcurrent,inisnominal inputhydrogenflow.

    Sincek/generallyarisesfromtheexternalload change,only?canberegulatedtomakethe

    constraintAu:0.Therefore,wemaykeep11withinits admissiblerangebyadjustingtheinputhydrogen flowrateaccordingto

    ?g

    ino

    

    ARXIDENTIFICATIONMODEL

    Amongallkindsofparametricmodels,theARX modelshouldbeselectedfirstlybecauseofitssim

    plicity.TheARXmodelhastheform:

    1923

()+qy(k1)+'?'+ana

    y(kHa)=b,u(k?Elk)+

    b2u(k一一1)+'?'+"(庀一一nh+1)+(),

    (10)

    orinthecompactformof

    A(q)y(k)

    A(q)

    B(q)

    B(q)"(

    1+lg?4-'

    4-b2q??'

    

    ,z)4-(),

    -.

    +a

    ng,a

    _+g,

    (11)

    (12)

    (13)

    where(and"(aretheprocessoutputandinputat timestep;gisthedelayoperator,e.g.,g"(= (庀一1);isawhitenoisesignal;Elkisthepure timedelayofthemodel,themodelorderofthe observedstate(alsocalledthenumberofpoles),FIb themodelorderofthecontrolsignal(alsocalledthe numberofzeros);afand6,aremodelparameters,with i=1,...,andj=1,...,6.

    FromEq.(10),itcanbeseenthattheARXmodel describestherelationshipamonginputsignals,output

    signalsandnoisethroughalineardifferenceequation, wheretheoutputatagiventimestepkiscomputedas alinearcombinationofpastoutputsandpastinputs. Duringtheidentification,allimportantstep,which precedestheparameterestimationoftheARXmodel showninEq.(10),istodeterminethemodelstructure, whichisentirelydefinedbythethreeintegersd,Elb andElk.Inthispaper.theoptimalARXmodelorderis selectedbyminimizingAkaikeinformationcriterion (AIC).Parameterestimationvialeastsquares(LS)

    withanARXmodelstructureisperhapsthemost widelyusedapproachtosystemidentification.Hence, afterthemodelstructureisdetermined,thestandard LSmethodisusedforonlineparameteridentification oftheARXmodel

    ITERATIVELEARNINGCONTROL

    AnILCisakindofcontrolalgorithmwhichis capableoftrackingadesiredtrajectoryperfectlyina period.Thisalgorithmsearchesanoptimalinput basedonpreviousresponsesbyiterations.Thebasic configurationofILCisillustratedinFig.2(Gopinath andKar,2004),where,"(denotestheinputsignal duringthefthiterationappliedtothesystem,Yi(the outputtrajectory,andyd(k)thedesiredtrajectory.

1924Huoetal/JZhejiangUnivSciA20078(12J:19211927

    Thesesignalsarestoredinthememoryuntiltheit

    erationiscompleted,atwhichtimetheyareprocessed ominebytheILCalgorithm.Thelearningcontroller isnothingbuttheILCalgorithmcomparesyd(k)with

    andaddstheupdatedterm(s)toui(toproduce Ui+l(k),therefinedinputsignalgiventothesystemfor the(i+1)thiteration.Theinputsignalui(keepsup

    datinguntiltherequirederrorgoalisreached. Fig.2Basiciterativelearningcontrolconfiguration Consideradiscretetime,linear,timeinvariant

    systemintheformusingZ-transferfunction: y(z)=,(z)(z)

    :

    (hz-m++Z-(re+1)++z+)(z),(I4)

    wheremistherelativedegreeofthesystem,zisthe

    standarddelayoperatorintime,andhiarethestan

    dardMarkovparametersofthesystem/4(z).Assume m=1.Ifthe"supervectors"(Mooreeta1.,2005)are definedasfollows:

    and

    Ui=[CO),),,(?一1)],

    =

    [Yi(1),(2),,(?)r,

    yd=[(1),Y(2),,YO(?)],

    (15)

    (16)

    (17)

    =

    [(1),(2),,(?)],(18)

    thenEq.(14)canbewrittenas

    Yi=H.U,(19)

    whereisamatrixofrankNwhoseelementsarethe Markovparametersoftheplant.And

    H.=

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